Control SystemsPreliminary engineering
Nyquist Plot & Stability Criterion
The L(jω) locus in the complex plane with the −1 point marked, plus the exact Nyquist accounting: open-loop RHP poles P, closed-loop RHP poles Z, and the implied encirclement count N.
About this calculator
The Nyquist criterion is the stability test that keeps working when Bode margins break down — when the open loop is already unstable, or the magnitude crosses 0 dB more than once. It counts encirclements of the critical point −1 by the open-loop locus L(jω): with P open-loop right-half-plane poles and N net clockwise encirclements of −1, the closed negative-unity-feedback loop has Z = N + P right-half-plane poles, and stability demands Z = 0.
This tool draws the positive-frequency locus exactly (the negative-frequency half is its mirror image, shown dashed) and marks −1. Rather than estimating the winding count off pixel geometry — fragile exactly where it matters — it computes the accounting algebraically: Z is the number of right-half-plane roots of the closed-loop characteristic polynomial D(s) + N(s), P comes from the open-loop poles, and N = Z − P follows. The plot is the geometric insight; the numbers are exact.
For type-1 and type-2 systems the locus runs off to infinity as ω → 0 — the classical infinite arc that closes the contour around the origin detour. The plot clips those samples at a stated radius so the finite geometry near −1 stays readable, and says so in a note rather than silently rescaling.
Distance from the locus to −1 is robustness itself: the gain and phase margins are just two one-dimensional slices of that distance, which is why a locus that skims the critical point warns you even when both margins look adequate.
Design notes & common mistakes
- The criterion counts encirclements of −1 (not the origin) because 1 + L = 0 is the characteristic equation — mapping the RHP boundary through L shifts the origin question to −1.
- Type ≥ 1 systems need the infinite-arc closure around the jω-axis detour at the origin: the locus closes through infinity by (type × 180°) clockwise. Forgetting the arc is the classic hand-sketch error; this tool's algebraic count is immune to it.
- A locus that passes THROUGH −1 is the marginal case (imaginary-axis closed-loop poles) — the count Z alone flags it here as boundary, not stable.
- Counterclockwise encirclements are negative N: an open-loop-unstable plant is stabilized precisely by making the locus wrap −1 counterclockwise P times.
Assumptions
- Negative unity feedback; the entered H(s) is the complete open-loop transfer function L(s).
- Z is computed exactly as the RHP root count of D(s) + N(s) rather than by graphical winding estimation; N is then implied by N = Z − P.
- The locus is sampled over the automatic frequency window (two decades beyond the extreme pole/zero magnitudes) — features outside it exist only for pathological coefficient spreads.
When to use this calculator
Appropriate for
- Applying the Nyquist criterion — especially when Bode margins are unreliable (open-loop RHP poles, multiple crossovers)
- Reading closed-loop stability from the encirclement accounting Z = N + P
- Teaching how the locus's approach to the −1 point relates to robustness
Not suitable for
- Systems with transport delay, whose true locus spirals in a way a rational plot cannot show
- MIMO systems, which need the generalized (eigenloci) Nyquist treatment
- Final stability certification on a real plant without validating the open-loop model
What this calculator does not cover
- The plotted locus is a finite-frequency sample with a stated clipping radius — the infinite arc of type ≥ 1 systems is described, not drawn.
- No transport delay e^(−sT); delay wraps the locus into a spiral that a rational H(s) cannot represent.
- Imaginary-axis OPEN-loop poles are handled by the algebraic count, but the textbook contour-indentation drawing around them is not rendered.
- SISO only — no MIMO generalized Nyquist / eigenloci.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
Why is the −1 point special?
Closed-loop poles solve 1 + L(s) = 0, i.e. L(s) = −1. The Nyquist locus is the image of the imaginary axis under L, so how it wraps around −1 counts exactly how many closed-loop poles fall in the right half-plane.
When do I need Nyquist instead of Bode margins?
Whenever the open loop has right-half-plane poles, when the magnitude crosses 0 dB multiple times, or when conditional stability is suspected (stable only within a band of gains). Bode margins assume away all three; the encirclement count does not.
How does this tool count encirclements without graphical tracing?
Algebraically: Z is the number of right-half-plane roots of D(s) + N(s), computed by the same tested root-finder used everywhere in this toolkit, and N follows from Z = N + P. The plot is for insight; the count never depends on plot resolution.
What happens to the plot for a system with an integrator?
As ω → 0 the locus magnitude grows without bound — the famous arc at infinity. The plot clips those samples at a stated radius so the region near −1 stays readable; the stability accounting is computed algebraically and is unaffected by the clipping.
References
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 10 (Nyquist criterion, encirclement counting)
- Franklin, G., Powell, J. & Emami-Naeini, A., Feedback Control of Dynamic Systems, 8th ed., Ch. 6 (Nyquist stability criterion)
- Ogata, K., Modern Control Engineering, 5th ed., Ch. 7 (Nyquist stability analysis)
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